1. Consider a consumer whose preferences can be represented by the utility function u(x; y) = x + y
(a) Originally, px = 1, py = 2 and m = 1. What bundle does the consumer choose, and what is his utility from this bundle?
(b) The price of good x then rises to 3. What bundle does the consumer choose after the price change, and what is his utility from this bundle?
(c) Calculate the compensating variation. (Hint: at the new price ratio, what good will he spend his income on?)
(d) Calculate the equivalent variation.
2. Consider a firm operating in a perfectly competitive market with the following production function:
f(x1; x2) =0.5*x1(1/2)*x2(1/4)
In the short run, x2 is fixed at 1 - the firm cannot increase or decrease its amount of this input, and must pay for this 1 unit of x2 regardless of what production decision it makes.
(a) Suppose that p, the price of the firm's output, is 1, and the input prices are w1 = 1 and w2 = 10. Calculate the firm's profit maximising choice of x1 and output.
(b) Calculate the firm's profit-maximising level of profit using your answers in part (a). You should see that the firm is making losses at its optimal choice of input and output. Explain in one sentence why the firm chooses to produce a positive output despite making losses.
(c) Derive the firm's short-run cost function and its short-run supply function (x2 is fixed at 1).
(d) Now suppose that x2 is still fixed at 1, but the firm's technology changes. Its new technology can be represented by the production function f(x1; x2) = x1(1/2)*x2(1/4) Without re-doing any calculations, how would you expect the firm's supply function to change relative to the supply you found in part a? Would it supply more output at any given price, less output, or the same amount of output, and why? Explain in no more than 2 sentences.